You don't have to let him cry - shush/pat if baby starts to cry after being put down. It will usually calm babies that young quickly. Certainly want to swaddle for naps and BT until 4 months, maybe longer depending on your LO. Mine is swaddled with one arm out at 4 1/2 months (naps and night).
It's okay to start the transition to crib with LO drowsy when put down, as opposed to fully awake
And remember that fussing and crying are different! There is also a mantra cry (I have never figured my baby's out though!) that will actually mean baby is trying to soothe himself and you doing anything during this cry will cause further problems for baby falling asleep. It's supposed to be a rhythmic sound - almost like a moaning type cry.